Pulsed Circuits

Abstract

In previous chapters, we exploited the symbolic method to study the behavior of electrical circuits with sinusoidal voltages and currents.

Problems

Problem 1

A: Obtain the differential equation for the output voltage $v_o(t)$ across the inductance L of the RL circuit shown in Fig. 9.11a when the input voltage is $v_i(t)$. B: Determine under what conditions the voltage $v_o(t )$ is proportional to the derivative of $v_i(t)$ in the case the input signal is a pulse of duration $\tau$. C: Solve the differential equation found in A in the case $v_i(t)$ is a voltage step function of amplitude V using appropriate initial conditions. [A. A: $dv_i(t)/dt=(R/L)v_L(t)+dv_L(t)/dt$; B: $(L/R)\ll \tau$; C: $v_L(t)=V\exp (-Rt/L)$.]

Problem 2

A: Obtain the differential equation for the output voltage $v_o(t)$ across the resistance R of the RL circuit shown in Fig. 9.11b when the input voltage is $v_i(t)$ B: Determine under what conditions the voltage $v_R(t )$ is proportional to the integral of $v_i(t)$ in the case the input signal is a pulse of duration $\tau$. C: Solve the differential equation found in A in the case $v_i(t)$ is a voltage step function of amplitude V using appropriate initial conditions. [A. A: $v_i(t)=(L/R)v_R(t)+dv_R(t)/dt$; B: $(L/R)\ll \tau$; C: $v_R(t)=V(1-\exp (-Rt/L)$).]

Problem 3

In the circuit shown in the figure the switch T is open at time $t=0$. Determine the voltage across the capacitance C as a function of time. [A. $v_C(t)=V_1+(V_2-V_1)\exp (-t/RC)$.]

Problem 4

Obtain the differential equation for the voltage v(t) across the capacitance C of the circuit shown in the figure. Solve the differential equation when $v_i$ is a voltage step function of amplitude V. [A. $v(t)=VR_2/(R_1+R_2)\exp (-t (R_1+R_2)/(R_1R_2C))$.]

Problem 5

Solve the previous problem using the Thévenin’s theorem.

Problem 6

In the circuit shown in the figure the switch T is closed at time $t=0$. Determine the voltage across the capacitance C as a function of time. [A. $v(t)=VR_3/(R_1+R_3)\exp (-t(R_3+R_1)/(R_3C(R_1+R_2+R_1R_2/R_3)))$.]

Problem 7

Solve the previous problem using the Thévenin’s theorem

Problem 8

Obtain the differential equation for the current $i_L(t)$ through the inductance L in the circuit shown in the figure, when $v_i(t)$ is the input voltage. [A. $v_i(t) R_3(R_1R_3+R_1R_2+R_2R_3)=i_L+L(R_1+R_3)/((R_1R_3+R_1R_2+R_2R_3)di_L/dt$.]

Problem 9

Solve the previous problem using the Norton’s theorem

Problem 10

Obtain the differential equation for the voltage $v_C(t)$ across the capacitance C in a RLC series circuit when $v_i(t)$ is the input voltage. Solve the differential equation, using the appropriate boundary conditions, when $v_i$ is a voltage step function of amplitude V. [A. $v_i(t)=v_C(t)+RCdv_C(t)/dt+LCd^2v_C(t)/dt^2$; $v_C(0)=0, dv_C/dt(0)=0$.]

Problem 11

As an alternative to the exponential pulse, a real voltage step can be described as a linear ramp of time duration T up to the constant value V. Using the convolution method, find the response of a high-pass RC circuit to this input waveform. Comment the solution obtained in terms of the behavior of the capacitor C. [A. $v_o(t) = 0\,\text {for}\,\,t<0, \quad v_o(t)=\alpha RC[1-\exp (-t/RC)]\,\,\, \text{ for }\,\,0 < t <T,v_o(t) = \alpha RC[1-\exp (-T/RC)] \exp [-(t-T)/RC] \quad \text{ for } t >T$.]

Problem 12

The circuit shown in the figure represents a real RL filter where the inductor L has a stray resistance $R_L$. Obtain the differential equation for the current $i_L(t)$ through the inductance L in response of an input voltage $v_i(t)$. Solve the differential equation when $v_i$ is a voltage step function of amplitude V. Finally obtain the output voltage $v_o(t)$ across the real inductor. [A. $v_o(t)=VR_L/(R+R_L)+VR/(R+R_L)exp(-t/\tau )$ with $\tau =L/(R+R_L)$.]

Problem 13

In the circuit shown in the figure the switch T is closed at time $t=0$. Obtain the differential equation for the current $i_L(t)$ in the inductance L as a function of time and determine the appropriate boundary conditions for the solution. [A. $V=R_1 i_L(t)+(L+R_1R_2C)di_L(t)/dt+R_1LCd^2i_L(t)/dt^2$ where $i_L(0)= V/(R_1+R_2)$, $di_L/dt(0)=V/L(R_1+R_2)$.]

Problem 14

The circuit in the figure is a diagram of the cable connection of a signal to an oscilloscope. Derive in time domain the differential equation yielding the signal $v_o(t)$ across the resistance $R_2$ when the input signal is $v_i(t)$. Find the appropriate boundary conditions for the solution when the input is a voltage step function. [A. $v_i(t)=(1+R_1/R_2)v_o(t)+(R_1C+L/R_2)dv_o(t)/dt+LCd^2v_o(t)/dt^2$; $v_o(0)=0,\, dv_o/dt(0)=0$.]

Problem 15

The circuit in the figure represents a filter LR realized with a real inductor whose series resistance is $R_L$ and whose parasitic capacitance is C. Obtain the differential equation that describes the response $v_o(t)$ to a signal $v_i(t)$. Find the appropriate boundary conditions in case the input signal is a step function of amplitude V. [A. $v_i(t)+RCdv_i(t)/dt+LCd^2v_i(t)/dt^2=(1+R_L/R)v_o(t)+(RC+L/R)dv_o(t)/dt+LCd^2v_o(t)/dt^2$; $v_o(0)=V, dv_o/dt(0)=-V/RC$.]